The geometric description of addition of points P and Q on an elliptic curve involves four logical branches: If one of P or Q is the point at infinity … Else if P = Q … Else if P and Q lie on a vertical line … Else … It would seem that an algorithm […]
Dimension 2 The equation for the perimeter of an ellipse is where a is the semimajor axis, e is eccentricity, and E is a special function. The equation is simple, in the sense that it has few terms, but it is not elementary, because it depends on an advanced function, the complete elliptic integral of the […]
When I shared an image from the previous post on Twitter, someone who goes by the handle Nonetheless made the astute observation that image looked like the Maxwell-Boltzmann distribution. That made me wonder what 1/Γ(x) would be like turned into a probability distribution, and whether it would be approximately like the Maxwell-Boltzmann distribution. (Here I’m […]
Thanks to the quadratic equation, it’s easy to tell whether a quadratic equation has a double root. The equation has a double root if and only if the discriminant is zero. The discriminant of a cubic is much less known, and the analogs for higher order polynomials are unheard of. There is a discriminant for […]
Given a field F, finite or infinite, you can construct a projective plane over F by starting with pairs of elements of F and adding “points at infinity,” one point for each direction. Motivation: Bézout’s theorem A few days ago I mentioned Bézout’s theorem as an example of a simple theorem that rests on complex […]
I’ve asserted numerous times here that hypergeometric functions come up very often in applied math, but I haven’t said why. This post will give one reason why. One way to classify functions is in terms of the differential equations they satisfy. Elementary functions satisfy simple differential equations. For example, the exponential function satisfies y′ = […]
I was thumbing through A New Kind of Science [1] and one of the examples that jumped out at me was looking at the bits in the binary representation of the powers of 3. I wanted to reproduce the image myself and here’s the result. Here a white square represents a 1 and a blue […]
This post muses about what it means to learn a software library. I’ll use Pandas as an example, but the post isn’t just about Pandas. Suppose you say “I want to learn Pandas.” That implicitly assumes Pandas one thing, and in a sense it is. In another sense Pandas is hundreds of things. At the […]
This post will look at how to take an expression for a Boolean function and look for a simpler expression that corresponds to the same function. We’ll show how to use a Python implementation of the Quine-McCluskey algorithm. Notation We will write AND like multiplication, OR like addition, and use primes for negation. For example, […]
In his book The Algorithm Design Manual, Steven Skiena has several sections called “War Stories” where he talks about his experience designing algorithms for clients. Here’s an excerpt of a story about finding the best airline ticket prices. “Look,” I said at the start of the first meeting. “This can’t be so hard. Consider a […]
